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Health System
RADIOLOGY RESEARCH
Henry Ford
NERS/BIOE 481
Lecture 02Radiation Physics
Michael Flynn, Adjunct Prof
Nuclear Engr & Rad. Science
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II.A – Properties of Materials (6 charts)
A) Properties of Materials
1) Atoms
2) Condensed media
3) Gases
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II.A.1 – Atoms
• The primary components of thenucleus are paired protons andneutrons.
• Because of the coulomb forcefrom the densely packedprotons, the most stableconfiguration often includesunpaired neutrons
Neutronsneutral charge1.008665 AMU
Protons+ charge1.007276 AMU
Electrons- charge0.0005486 AMU
Positrons+ charge0.0005486 AMU
C136
Carbon 1313 nucleons6 protons7 neutrons
-
+
Terminology
���
A = no. nucleons
Z = no. protons
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II.A.1 – Atoms – the Bohr model
• The Bohr model of the atomexplains most radiation imagingphenomena.
• Electrons are described asbeing in orbiting shells:
• K shell: up to 2 e-, n=1
• L shell: up to 8 e-, n=2
• M shell: up to 18 e-, n=3
• N shell: up to 32 e-, n=4
• The first, or K, shell is the mosttightly bound with the smallestradius. The binding energy(Ionization energy in eV)neglecting screening is ……. )( 22
0 nZII
C136-
--
-
-
-
eV
Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part I“, Philosophical Magazine 26 (151): 1–24.
Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part II“, Philosophical Magazine 26 (153): 476-502.
Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part III“, Philosophical Magazine 26 (155): 857-875.
60.130 I
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II.A.2 – Condensed media
• For condensed material, themolecules per cubic cm can bepredicted from Avogadro’snumber (atoms/cc)
ANN am
• Consider copper with oneatom per molecule,
A = 63.50
r = 8.94 gms/cc
Ncu = 8.47 x 1022 #/cc
• If we assume that the copper atomsare arranged in a regular array, we candetermine the approximate distancebetween copper atoms (cm);
8
31 103.21
Cu
CuN
lcm
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II.A.2 – radius of the atom
•The Angstrom originated as a unit appropriate fordescribing processes associated with atomicdimensions. 1.0 Angstroms is equal to 10-8 cm. Thus theapproximate spacing of Cu atoms is 2.3 Angstroms.
• In relation, the radius of the outer shell electrons(M shell) for copper can be deduced from theunscreened Bohr relationship (Angstroms);
Angstromsr
r
Znr
Cu
Cu
Hm
,16.
29352917. 2
2
Thus for this model of
copper, the atoms constitutea small fraction of the space,
VCu = (4/3)p0.163 = .017 A3
VCu / (2.33) = .001
aH is the ‘Bohr radius’, the radius ofthe ground state electron for Z = 1
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II.A.3 – the ideal gas law
• An ideal gas is defined as one in
which all collisions between atoms
or molecules are perfectly elastic
and in which there are no
intermolecular attractive forces.
One can visualize it as a collection
of perfectly hard spheres which
collide but which otherwise do not
interact with each other.
• An ideal gas can be characterized
by three state variables: absolute
pressure (P), volume (V), and
absolute temperature (T). The
relationship between them may be
deduced from kinetic theory and is
called the “ideal gas law”.
NkTnRTPV P = pressure, pascals(N/m2)
V = volume, m3
n = number of moles
T = temperature, Kelvin
R = universal gas constant
= 8.3145 J/mol.K(N.m/mol.K)
N = number of molecules
k = Boltzmann constant
= 1.38066 x 10-23 J/K
= 8.617385 x 10-5 eV/K
k = R/NA
NA = Avogadro's number
= 6.0221 x 1023 /mol
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II.A.3 – air density
•The density of a gas canbe determined bydividing both sides ofthe gas equation by themass of gas contained inthe volume V.
r = (P/T)/Rg
•The gas constant for aspecific gas, Rg, is theuniversal gas constantdivided by the grams permole, m/n.
Rg = R / (m/n)
• m/n is the atomic weight.
TRP
TRm
n
m
VP
nRTPV
g
Dry Air example
Molar weight of dry air = 28.9645 g/mol
Rair = (8.3145/28.9645) =.287 J/g.K
Pressure = 101325 Pa (1 torr, 760 mmHg)
Temperature = 293.15 K (20 C)
Density = 1204 (g/m3) = .001204 g/cm3
Note: these are standard temperature andpressure, STP, conditions.
gRRm
n
Rg = specific gas constant
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II.B – Properties of Radiation (3 charts)
B) Properties of Radiation
1) EM Radiation
2) Electrons
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II.B.1 – EM radiation
• The electric and magnetic fields are perpendicular to eachother and to the direction of propagation.
• X-ray and gamma rays are both EM waves (photons)
• Xrays – produced by atomic processes
• Gamma rays – produced by nuclear processes
• The energy of an EM radiation wave packet (photon) isrelated to the wavelength;
• E = 12.4/l , for E in keV & l in Angstrom
• E = 1.24/l , for E in keV & l in nm
• E = 1240/l , for E in eV & l in nm
Electromagnetic radiationinvolves electric andmagnetic fields oscillatingwith a characteristicfrequency (cycles/sec)and propagating in spacewith the speed of light.
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II.B.1 – EM radiation
The electromagnetic spectrum covers a wide range of wavelengths
and photon energies. Light used to "see" an object must have a
wavelength about the same size as or smaller than the object.
Lawrence Berkeley Lab: www.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html
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II.B.2 – electron properties
• The electron is one of a class of subatomic particlescalled leptons which are believed to be “elementaryparticles”. The word "particle" is somewhatmisleading however, because quantum mechanicsshows that electrons also behave like a wave.
• The antiparticle of an electron is the positron,which has the same mass but positive rather thannegative charge.
http://en.wikipedia.org/wiki/Electron
•Mass-energy equivalence = 511 keV
•Molar mass = 5.486 x 10-4 g/mol
•Charge = 1.602 x 10-19 coulombs
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II.C – Radiation Interactions (C.1 7 charts)
C) Radiation Interactions
1) Electrons
2) Photons
a. Interaction cross sections
b. Photoelectric interactions
c. Compton scattering (incoherent)
d. Rayleigh scattering (coherent)
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II.C.1 – Electron interactions
Basic interactions of electrons and positrons with matter.
Ee
Elastic Scattering
Ee
Ee
Bremsstrahlung (radiative)
W
E - W
Ee
Inelastic Scattering
Ee - W
W - Ui
Ep
Positron Annihilation
511 keV
511 keV
++-X
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PENELOPE
• Tungsten
• 10mm x 10mm
• 100 keV
For take-offangles of
17.5o-22.5o
0.0006 of theelectronsproduce anemitted x-rayof some energy.
Numerous elasticand inelasticdeflections causethe electron totravel in atortuous path.
II.C.1 – Electron multiple scattering
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II.C.1 – Electron path
• A very large number of interactions with typicallysmall energy transfer cause gradual energy loss asthe electron travels along the path of travel.
1 10 100 10001
10Molybdenum(42)
Tungsten(74)
Me
V/(
g/c
m2)
keV
Electron Stopping Power
http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html
•The ContinuousSlowing DownApproximation (CSDA)describes the averageloss of energy oversmall path segments.
• ICRU reports 37 (1984)and 49 (1993).
• Berger & Seltzer, NBS82-2550A, 1983.
• Bethe, Ann. Phys., 1930
~E-0.65
MeV/(g/cm2) - For radiation interaction data, units of distance are often scaled usingthe material density, distance * density, to obtain units of g/cm2.
dE/ds
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10 100
1E-3
0.01
0.1
Molybdenum(42)
Tungsten(74)
CSDA Pathlength
path
len
gth
/de
nsit
y(g
m/c
m2)
keV
II.C.1 – Electron pathlength (CSDA)
The total pathlength traveled by the electron along thepath of travel is obtained by integrating the inverse ofthe stopping power, i.e. 1/(dE/ds) ,
gm/cm2
Pathlength is oftennormalized as theproduct of the length incm and the materialdensity in gm/cm3 toobtain gm/cm2.
CSDA –
Continuous SlowingDown Approximation.
dE
dsdE
RT
CSDA 0 1
• 100 keV, Tungsten, 15.4 mm
• 30 keV, Molybdemun, 3.2 mm
~E1.63
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II.C.1 – Electron transport
• A beam of many electrons striking a target willdiffuse into various regions of the material.
50 e
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• Electrons are broadlydistributed in depth, Z,as they slow down dueto extensive scattering.
• Most electrons tend torapidly travel to themean depth and diffusefrom that depth.
Electron Depth Distribution100 keV Tunsten(Penelope MC)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
microns
P(z
)dz
50 keV
60 keV
70 keV
80 keV
90 keV
Pz(T,Z) is thedifferential probability(1/cm) of that anelectron within thetarget is at depth Z
II.C.1 – Electron depth distribution vs T
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II.C.1 – Electron extrapolated range
The extrapolated range is commonly measured from electrontransmission data measured with foils of varying thickness. It isdefined as the point where the tangent at the steepest point on thealmost straight descending portion of the penetration curve.
21 /eR gm cm
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II.C.1 – Electron extrapolated range
• The extrapolated range in units of gm/cm2is nearly independent of atomic number.
• The extrapolated range is about 30-40%of the CSDA pathlength.
Tabata, NIM Phys. Res. B, 1996
10 1001E-4
1E-3
0.01
Molybdenum(42)
Tungsten(74)
Extrapolated Range
ran
ge
/den
sit
y(g
m/c
m2 )
keV10 100
0
1
Molybdenum(42)
Tungsten(74)
Extrapolated rangeas a fraction of
CSDA Pathlength
Ra
ng
e/P
ath
len
gth
keV
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II.C – Radiation Interactions (C.2 18 charts)
C) Radiation Interactions
1) Electrons
2) Photons
a. Interaction cross sections
b. Photoelectric interactions
c. Compton scattering (incoherent)
d. Rayleigh scattering (coherent)
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II.C.2.a – Cross sections
• The probability of a specificinteraction per atom is known
as the cross section, s.
• The probability is expressed asan effective area per atom.
• The ‘barn’ is a unit of areaequal to 10-24 cm2 (non SI unit).
• The probability per unit thickness that aninteraction will occur is the product of the crosssection and the number of atoms per unit volume.
μ = s N , (cm2/atom).(atoms/cm3) = cm-1
N = Na (r /A) Na - Avogadro’s # (6.022 x 1023)r - densityA - Atomic Weight
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II.C.2.a – Cross sections for specific materials
• Cross section values aretabulated for theelements and manycommon materials.
• Values range from 101-104 barns (10 to 100keV) depending on Z.
• Cross sections aretypically small relativeto the area of the atom.
Molybdenum
MeV
Barnsatom
http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html
Cu, 30 keV
s = 1.15 x 103 b/atom
ACu = 8.04 x 106 b/atom
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II.C.2.a – Linear attenuation coefficients
• The probability per unitthickness that an x-ray willinteract when traveling asmall distance called the‘linear attenuationcoefficient’.
• For a beam of x-rays, therelative change in thenumber of x-rays isproportional to the incidentnumber.
• For a thick object ofdimension x, the solution tothis differential equation isan exponential expressionknown as Beer’s law.
NdX
dN
X
N
DX
N - DNN
xx eNN )0()(
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II.C.2.a – Xray Interaction types
Attenuation is computed from Beer’s law using the linear attenuation
coefficient, m, computed from cross sections and material composition.
Photoelectric effect Coherent scatter
μPE = N1σ 1PE + N2σ 2PE + N3σ 3PE
Where Ni is the number of atoms of type i
and si is the cross section for type i atoms.
xo eNN
In the energy range of interest for diagnostic xray imaging, 10 – 250 keV,there are three interaction processes that contribute to attenuation.
μ = μPE + μINC + μCOH
Incoherent scatter
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II.C.2.b – Photoelectric Absorption
Photoelectric Absorption
• The incident photon transfers all ofit’s energy to an orbital electron anddisappears.
• The photon energy must be greaterthan the binding energy, I, forinteraction with a particular shell.This cause discontinuities inabsorption at the various I values.
• An energetic electron emerges fromthe atom;
Ee = Eg – I• Interaction is most probable for the
most tightly bound electron. A K shellinteraction is 4-5 times more likelythan an L shell interaction.
• Very strong dependance on Z and E.
Photoelectric effect
3
4
E
Z
Akpe
60.13
)(
0
220
I
nZII
eV
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II.C.2.b – Photoelectric Cross Section vs E
Atomic photoelectric cross sections for carbon, ironand uranium as functions of the photon energy E.
from Penelope, NEA 2003 workshop proceedings
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II.C.2.c – Compton Scattering
Compton Scattering
• An incident photon of
energy Eg interactswith an electron with areduction in energy, E’g,and change in direction.(i.e. incoherent scattering)
• The electron recoilsfrom the interaction inan opposite direction
with energy Ee.
Incoherent scatter
-Eg
E’g
Ee
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II.C.2.c – Compton Scattering – energy transfer
The scattered energyas a fraction ofincident energydepends on the angleand scaled energy, a.
Barrett, pg 322
Incident Energy, Eg , MeV
(E’g / Eg)
)(511
cos1111
2
2
keVcm
cm
E
EEE
o
o
cos11
1
E
E
-
q
fEg
E’g
Ee
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Since Z/A is about 0.5 for all but verylow Z elements, the mass attenuation
coeffient, mc/r , is essentially thesame for all materials !
II.C.2.c – Compton Scattering – cross section
• The cross section for compton scattering is expressed as theprobability per electron such that the attenutation coefficientfor removal of photons from the primary beam is;
co
ce
c
A
ZNn
• The Klein-Nishina equation describes thecompton scattering cross section in relationto a classical cross section for photonscattering that is independent of energy(so , Thomson free electron cross section).
)( KNoc f
222
2
21
311
2
1)21ln(
21
12
4
3)(
KNf
Barret, App. C, pg 323
so=(8p /3) re2
re=e2/(moc2)
so= 0.6652 barn
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II.C.2.c – Compton Scattering – cross section vs E
• The free electroncompton scatteringcross section is slowlyvarying with energy forlow photon energies.
• At high energy, a > 1,the cross section is
proportional to 1/E.
Barret, App. C, pg 323
E, MeV
cm2
• The total scattering cross section is made of twocomponent probabilities, sen and ss, describing how muchof the photon energy is absorbed by recoil electrons andhow much by the scattered photon. This difference isimportant in computations of dose and exposure.
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II.C.2.c – Compton Scattering – cross section vs E
• The differential scatteringcross section describes theprobability that the scatteredphoton will be deflected into athe differential solid angle, dW,in the direction (q,f).
• The cross section is forwardpeaked at high energies.
Barret, App. C, pg 326
In the next lecture, we willlearn more about solid angleintegrals. Since the differentialcross for unpolarized photonsdepends only on q, we will find,
dd
d cc sin2
0
d
d c
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II.C.2.c – Compton vs Photoelectric
Photoelectric
Dominant forhigh Z at lowenergy.
Compton
Dominant for lowZ at mediumenergy.
Pair production
Dominant forhigh Z at veryhigh energy.
0.01 0.10 1.00 10.0 100
100
80
60
40
20
0
COMPTON
PHOTO PAIR
Z
MeV
In the field of a nucleus,a photon may annihilatewith the creation of anelectron and a positron.
Eg > 2 x 511 keV
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II.C.2d – Rayleigh Scattering
Rayleigh Scattering
• An incident photon of
energy Eg interactswith atomic electronswith a change indirection but noreduction in energy.(i.e. coherent scattering)
Coherent scatter
Eg
Eg
The Thomson cross section can be understood as a dipole interaction of anelectromagnetic wave with a stationary free electron. The electric field of theincident wave (photon) accelerates the charged particle which emits radiationat the same frequency as the incident wave. Thus the photon is scattered.
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II.C.2d – Rayleigh ScatteringIn the energy range from 20 – 150 keV,coherent scattering contributes 10% - 2% tothe water total attenuation coefficient.
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1 10 100 1,000keV
Xray Attenuation Coefficients for Water (cm2/g XCOM)
Rayleigh Scattering (COH)
Compton Scattering (INC)
Photoelectric Absorption
Total (with COH)
20 to 150 keV
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II.C.2d – Rayleigh Scattering
Rayleigh Scattering
• Rayleigh scattering is the coherent interaction of photons with thebound electrons in an atom. The angular dependence is described by
the differential cross section, dsR/dW.
• It is common to consider coherent scattering as a modification, withan atomic form factor, F(c, Z) , of the Thomson cross section,
• For the differential Thomson cross section per electron,
• It depends on the classical electron radius, re2 = 0.079b (slide 30).
• When integrated, the total cross section is the same as so (slide 30).
• The angular distribution is the same as for low energy Compton scattering (slide 32).
• The total cross section is then given by,
Even though there are other components to the totalcoherent scattering, such as nuclear Thomson, Delbruck,and resonant nuclear scattering, Rayleigh scattering isthe only significant coherent event for keV photons.
� � �� Ω
= � � , � �� � � �� Ω
� = sin1
2� ��
� � � �� Ω
=1
2� �� 1 + cos � �
� � = � � �� � � � , � �
�
� �
1 + cos � � � cos �
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II.C.2d – Rayleigh Scattering
� � = � � �� � � � , � �
�
� �1 + cos � � � cos �
At low values of c (small angle, low energy)the form factor is equal to the number ofelectrons (i.e 6 for Carbon in this example).
0.0
0.1
0.1
0.2
0.2
0.3
0.00 10.00 20.00 30.00 40.00 50.00 60.00Theta, degrees
Differential Rayleigh Scattering
dsR/dW= F2(X,Z) dsTh/dWHubbel, J. Phys. Chem. Ref. Data, 1979
6C
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II.C.2d – Rayleigh Scattering
When plotted vs angle, q ,coherent scattering is seento be very forward peaked.
E = 62 keV
l = 0.2 Angstroms
II.C.2d – Rayleigh Scattering
• Coherent scatteringfrom molecules andcompounds is morecomplex that foratoms.
• King2011 – King BW et. al. ,Phys. Med. Biol, 56, 2011.
• This has beeninvestigated as a wayto obtain materialspecific images.
• Westmore1997 – WestmoreMS et.al., Med.Phys,24,1997
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II.D – Radiation Interactions (8 charts)
D) X-ray generation
1) Atoms and state transitions
2) Bremmsstralung production
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II.D.1 – Atomic levels
•Each atomic electron occupies a single-particle orbital, withwell defined ionization energy.
•The orbitals with the same principal and total angularmomentum quantum numbers and the same parity make a shell.
C136-
--
-
-
-
Each shellhas a finitenumber ofelectrons,withionizationenergy Ui.
Ka2 , Ka1, Kb2 , Kb1,
from Penelope, NEA 2003 workshop proceedings
a2 a1 b3 b1 b2
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II.D.1 – Atomic relaxation
• Excited ions with a vacancy in aninner shell relax to their groundstate through a sequence ofradiative and non-radiativetransitions.
• In a radiative transition, thevacancy is filled by an electron froman outer shell and an x ray withcharacteristic energy is emitted.
• In a non-radiative transition, thevacancy is filled by an outerelectron and the excess energy isreleased through emission of anelectron from a shell that is fartherout (Auger effect).
• Each non-radiative transitiongenerates an additional vacancy thatin turn, migrates “outwards”.
Radiative
Auger
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II.D.1 – Fluorescent Fraction
Relative probabilities for radiative and Augertransitions that fill a vacancy in the K-shell of atoms.
from Penelope, NEA 2003 workshop proceedings
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II.D.2 – Bremsstralung
• In a bremsstrahlung event, acharged particle with kineticenergy T generates a photonof energy E, with a value inthe interval from 0 to T.
T
Bremsstrahlung (radiative)
E
T- E
• Bremsstrahlung (braking radiation) production results from thestong electrostatic force between the nucleus and the incidentcharged particle.
• The acceleration produced by a nucleus of charge Ze on a
particle of charge ze and mass M is proportional to Zze2/M.Thus the intensity (i.e. the square of the amplitude) will vary as
Z2z2/M2
• For the same energy, protons and alpha particles produce about10-6 as much bremsstrahlung radiation as an electron.
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II.D.2 – Brems. Differential Cross Section (DCS)
• The probability per atom that anelectron traveling with energy T willproduce an x-ray within the energyrange from E to E+dE is known as theradiative differential cross section
(DCS) , dsr/dE.
• Theoretic expressions indicate thatthe bremsstrahlung DCS can beexpressed as;
• Where b is the velocity of theelectron in relation to the speed oflight.
• The slowing varying function,fr(T,E,Z), is often tabulated as thescaled bremsstrahlung DCS.
E
Zf
dE
dZETr
r 12
2,,
e-
EIncidentenergy To
Energy T
Seltzer SM & Berger MJ,
Atomic Data & Nucl. Data Tables,
35, 345-418(1986).
2
22
)1(
11
cmT
e
E
Zf
dE
dZETr
r 12
2
),,(
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II.D.2 – scaled bremsstrahlung DCS
Seltzer and Berger, 1986, from Penelope, NEA 2003 workshop proceedings
Numerical scaled bremsstrahlung energy-loss DCS of Al and Au asa function of x-ray energy relative to electron energy, W/E (E/T).
(mba
rns)
(mba
rns)
Corrected – barns -> mbarns ( i.e. = E/T ) ( i.e. = E/T )
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II.D.2 – angular dependence of DCS
srQ - Barns/nuclei/keV/sr
sr - Barns/nuclei/keV
Q - electron (f,q) – xray (a)
a - xray takeoff angle
(f,q) – electron vector direction
e-E
T =0
To
Q
a
(f,q)
For convenience, the radiative DCS iswritten as,
A doubly differential cross section is usedfor the interaction probability differential inx-ray energy, E, and solid angle, W , in thedirection q ,
The shape function for atomic-fieldbremsstrahlung is defined as the ratio ofthe cross section differential in photonenergy and angle to the cross sectiondifferential only in energy.
And thus,
� � , � , Θ =� � �� �
� � � = � � , � ,Θ � �
� � ≡� � �
� ��
� � � ≡� � � �
� � � Ω�
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II.D.2 – Kissel shape function, S(T,E)
Brems Shape Function100 keV electrons
Molybdenum
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 20 40 60 80 100 120 140 160 180
Angle (degrees)
S-
(1/s
r)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
1
1/4pi
Eg / T
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II.D – Emission of Gamma radiation
In lecture 04 will consider thenuclear processes associated withthe generation of gamma radiation.
a. n/p stability
b. beta emission
c. electron capture
d. positron emission
